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mathematics
advanced engineering mathematics
Questions and Answers of
Advanced Engineering Mathematics
Is the given sequence z1, z2, · · ·, zn, · · · bounded?Convergent? Find its limit points. Show your work in detail.zn = nπ/(4 + 2ni)
Find the Maclaurin series and its radius of convergence.sin 2z2
Prove that n√n → 1 as n → ∞, as claimed.
What do you know about convergence of power series?
Find the Maclaurin series and its radius of convergence.
Show that (a) By using the Cauchy product. (b) By differentiating a suitable series.
Where does the power series converge uniformly? Give reason.
What is a Taylor series? Give some basic examples.
Show that if ∑αnzn has radius of convergence R (assumed finite), then ∑αnzn has radius of convergence √R.
Is the given sequence z1, z2, · · ·, zn, · · · bounded?Convergent? Find its limit points. Show your work in detail.zn = (-1)n + 10i
Find the Maclaurin series and its radius of convergence.
Find the radius of convergence in two ways: (a) Directly by the Cauchy–Hadamard formula in Sec. 15.2. (b) From a series of simpler terms by using Theorem 3 or Theorem 4.
What do you know about adding and multiplying power series?
Find the Maclaurin series and its radius of convergence.
Where does the power series converge uniformly? Give reason.
Find the Maclaurin series and its radius of convergence.cos2 1/2z
Find the center and the radius of convergence.
Does every function have a Taylor series development? Explain.
Is the given sequence z1, z2, · · ·, zn, · · · bounded?Convergent? Find its limit points. Show your work in detail.zn = n2 + i/n2
Can properties of functions be discovered from Maclaurin series? Give examples.
Find the Maclaurin series and its radius of convergence.sin2 z
Where does the power series converge uniformly? Give reason.
What do you know about term wise integration of series?
Find the center and the radius of convergence.
Is the given sequence z1, z2, · · ·, zn, · · · bounded?Convergent? Find its limit points. Show your work in detail.zn = (3 + 3i)-n
Find the Maclaurin series and its radius of convergence.
Find the radius of convergence in two ways: (a) Directly by the Cauchy–Hadamard formula in Sec. 15.2. (b) From a series of simpler terms by using Theorem 3 or Theorem 4.
How did we obtain Taylor’s formula from Cauchy’s formula?
Prove that the series converges uniformly in the indicated region.
Find the radius of convergence.
Find the center and the radius of convergence.
Write a program for graphing complex sequences. Use the program to discover sequences that have interesting “geometric” properties, e.g., lying on an ellipse, spiraling to its limit, having
Find the Maclaurin series by term wise integrating the integrand. (The integrals cannot be evaluated by the usual methods of calculus. They define the error function erf z, sine integral Si(z), and
Find the radius of convergence in two ways: (a) Directly by the Cauchy–Hadamard formula in Sec. 15.2. (b) From a series of simpler terms by using Theorem 3 or Theorem 4.
Find the Maclaurin series by term wise integrating the integrand. (The integrals cannot be evaluated by the usual methods of calculus. They define the error function erf z, sine integral Si(z), and
Prove that the series converges uniformly in the indicated region.
Find the radius of convergence.
Find the center and the radius of convergence.
Show that a complex sequence is bounded if and only if the two corresponding sequences of the real parts and of the imaginary parts are bounded.
Find the Maclaurin series by term wise integrating the integrand. (The integrals cannot be evaluated by the usual methods of calculus. They define the error function erf z, sine integral Si(z), and
Find the radius of convergence in two ways: (a) Directly by the Cauchy–Hadamard formula in Sec. 15.2. (b) From a series of simpler terms by using Theorem 3 or Theorem 4.
Find the Maclaurin series by term wise integrating the integrand. (The integrals cannot be evaluated by the usual methods of calculus. They define the error function erf z, sine integral Si(z), and
Prove that the series converges uniformly in the indicated region.
Find the radius of convergence.
Find the center and the radius of convergence.
(a) The Maclaurin series Show that E0 = 1, E2 = -1, E4 = 5, E6 = -61. Write a program that computes the E2n from the coefficient formula in (1) or extracts them as a list from the series. (b) The
Find the radius of convergence in two ways: (a) Directly by the Cauchy–Hadamard formula in Sec. 15.2. (b) From a series of simpler terms by using Theorem 3 or Theorem 4.
Clearly, from series we can compute function values. In this project we show that properties of functions can often be discovered from their Taylor or Maclaurin series. Using suitable series, prove
Prove that the series converges uniformly in the indicated region.
Find the radius of convergence. Try to identify the sum of the series as a familiar function.
Find the center and the radius of convergence.
Is the given series convergent or divergent? Give a reason. Show details.
State clearly and explicitly where and how you are using Theorem 2.If in (2) is odd (i.e., f(-z) = -f(z)), show that αn = 0 for even n. Give examples.
Find the Taylor series with center z0 and its radius of convergence.1/(1 - z), z0 = i
Find the radius of convergence. Try to identify the sum of the series as a familiar function.
Write a program for computing R from (6), (6*), or (6**), in this order, depending on the existence of the limits needed. Test the program on some series of your choice such that all three formulas
Is the given series convergent or divergent? Give a reason. Show details.
Show that (9) in Sec. 12.6 with coefficients (10) is a solution of the heat equation for t > 0 assuming that f(x) is continuous on the interval 0 ≤ x ≤ L and has one-sided derivatives at all
Find the radius of convergence. Try to identify the sum of the series as a familiar function.
Find the Taylor series with center z0 and its radius of convergence.sin z, z0 = π/2
Is the given series convergent or divergent? Give a reason. Show details.
Find the Taylor series with center z0 and its radius of convergence.cosh (z - πi), z0 = πi
Find the Maclaurin series and its radius of convergence. Show details.(sinh z2)/z2
Find the Maclaurin series and its radius of convergence. Show details.1/(1 - z)3
Find the Taylor series with center z0 and its radius of convergence.1/(z + i)2, z0 = i
Find the Maclaurin series and its radius of convergence. Show details.cos2 z
Is the given series convergent or divergent? Give a reason. Show details.
Find the Maclaurin series and its radius of convergence. Show details.1/(πz + 1)
Find the Taylor series with center z0 and its radius of convergence.sinh (2z - i), z0 = i/2
Find the Maclaurin series and its radius of convergence. Show details.-(exp/(-z2) - 1)/z2
Is the given series convergent or divergent? Give a reason. Show details.
Find the Taylor series with the given point as enter and its radius of convergence.cos z, 1/2π
Write a program for computing and graphing numeric values of the first n partial sums of a series of complex numbers. Use the program to experiment with the rapidity of convergence of series of your
Find the Taylor series with the given point as enter and its radius of convergence.Ln z, 3
Show that if a series converges absolutely, it is convergent.
Find the Taylor series with the given point as enter and its radius of convergence.ez, πi
Determine the location and order of the zeros.sin4 1/2z
One “rectangle” and its image are colored. Identify the images for the other “rectangles.”
If z moves from z = 1/4 twice around the circle |z| = 1/4, what does w = √z do?
Find the image of x = c = const, -π < y ≤ π, under w = ez.
Expand the function in a Laurent series that converges for 0 < |Z| < R and determine the precise region of convergence. Show the details of your work.cos z/z4
Evaluate the following integrals and show the details of your work.
What is a Laurent series? Its principal part? Its use? Give simple examples.
Expand the function in a Laurent series that converges for 0
What kind of singularities did we discuss? Give definitions and examples.
Determine the location and order of the zeros.(z4 + 81i)4
Draw an analog of Fig. 378 for w = z3.
Make a sketch, similar to Fig. 395, of the Riemann surface of w = 4√z + 1.
If you are familiar with 2 × 2 matrices, prove that the coefficient matrices of (1) and (4) are inverses of each other, provided that αd - bc = 1, and that the composition of LFTs corresponds to
Find and sketch the image of the given region under w = ez.-1/2 ≤ x ≤ 1/2, -π ≤ y ≤ π
Expand the function in a Laurent series that converges for 0 < |Z| < R and determine the precise region of convergence. Show the details of your work.exp z2/z3
Find all the singularities in the finite plane and the corresponding residues. Show the details.sin 2z/z6
Evaluate the following integrals and show the details of your work.
What is the residue? Its role in integration? Explain methods to obtain it.
Can the residue at a singularity be zero? At a simple pole? Give reason.
Evaluate the following integrals and show the details of your work.
What is an entire function? Can it be analytic at infinity? Explain the definitions.
Write a program for calculating the residue at a pole of any order in the finite plane. Use it for solving Probs. 5–10.Data from Prob. 5Find all the singularities in the finite plane and the
Find the images of the lines x = c = const under the mapping w = cos z.
Evaluate (counterclockwise). Show the details.
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